Naurgul
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 Sep30 comment How to intercept someone moving in a 2-dimensional grid world? Thanks, I was kinda hoping for an easier method though. Oh well. By the way, I updated the question to include some details of $d$'s motion to help motivate the question and help you make sense of what I'm trying to achieve. Sep30 comment How to intercept someone moving in a 2-dimensional grid world? @newbie Yes, every person can pick at most one direction and move at most one square in that direction. Nov20 comment To operate or not to operate? About the functions being monotonically increasing: I meant that each day you have an independent chance of dying. The same way that you can throw a die 100 times and still have a 1/6 chance that you get a six each time. About expectation values: The question is about finding the best day to have the operation, in order to maximise the expected lifespan. In my proposed solution, I didn't think it necessary to go through expected values calculations to achieve that. Why should it be necessary anyway?